The trilogy, comprising Not I, Rockaby and Footfalls, runs at the Court from 9 to 18 January 2014, and will play a two-week run at the Duchess from 3 to 15 February.
All tickets for the West End run will be sold at Royal Court prices (£12-£25).
Directed by Beckett's long-time collaborator, Walter Asmus, the trilogy will tour to Cambridge Arts Theatre, Birmingham Repertory Theatre and The Lowry, Salford in autumn 2014, with international dates to be added.
Royal Court artistic director Vicky Featherstone said: "In a visionary move and one which reminds us there is passion and integrity in our West End, Nica Burns and Nimax are opening up this extraordinary piece of theatre to a wider audience. That they are doing this and keeping tickets at Royal Court prices for a limited residency, makes me deeply proud and inspired to be in partnership with them."
Nimax's chief executive Nica Burns responded: "I'm delighted to enable all the audiences who could not get tickets at the Royal Court to experience Lisa Dwan's outstanding performance of these wonderful Beckett short plays. The delightfully intimate and atmospheric Duchess Theatre could not be a more perfect space to see the plays in the West End."
Also at the Court, it's been announced that Saskia Reeves and Danny Webb will play 'She' and 'He' in The Mistress Contract, a new play by Abi Morgan, which opens in the Jerwood Theatre Downstairs on 5 February (previews from 30 January).
The production, which is directed by Vicky Featherstone, has been extended for two weeks and will now run until 22 March.
Based on tape recordings made by a real couple during their 30-year relationship, The Mistress Contract is a billed as "a remarkable document of this unconventional couple, and the contract that kept them bound together to this day".
Saskia Reeves last appeared at the Royal Court in 2005 in The Woman Before. Her other theatre credits include Hello and Goodbye at Trafalgar Studios and A Disappearing Number for Complicité at the Barbican.
Danny Webb most recently appeared in the Royal Court's production of Circle Mirror Transformation in Haggerston, and in The Witness and Chicken Soup with Barley. Credits elsewhere include Troilus and Cressida at the RSC, Blasted at the Lyric Hammersmith and 13 at the National Theatre.